Indefinite theta functions arising from affine Lie superalgebras and sums of triangular numbers

TL;DR

This paper extends the theory of indefinite theta functions to prove modularity of certain power series, deriving identities for triangular numbers and linking them to affine Lie superalgebra denominator identities.

Contribution

It introduces new connections between indefinite theta functions, modular forms, and affine Lie superalgebras, expanding the theoretical framework and providing new identities.

Findings

Power series are proven to be modular forms.

Derived identities for powers of the triangular number generating function.

Identified specializations of affine Lie superalgebra denominator identities.

Abstract

We extend the recently developed theory of Roehrig and Zwegers on indefinite theta functions to prove certain power series are modular forms. As a consequence, we obtain several power series identities for powers of the generating function of triangular numbers. We also show that these identities arise as specializations of denominator identities of affine Lie superalgebras.